@article{LieSullivan2018, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013)}, volume = {34}, journal = {Inverse Problems}, number = {12}, doi = {10.1088/1361-6420/aae55b}, pages = {129601}, year = {2018}, language = {en} } @article{OatesSullivan2019, author = {Oates, Chris and Sullivan, T. J.}, title = {A modern retrospective on probabilistic numerics}, volume = {29}, journal = {Statistics and Computing}, number = {6}, arxiv = {http://arxiv.org/abs/1901.04457}, doi = {10.1007/s11222-019-09902-z}, pages = {1335 -- 1351}, year = {2019}, abstract = {This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related to modern formal treatments and applications. We highlight in particular the parallel contributions of Sul'din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.}, language = {en} } @misc{OatesCockaynePrangleetal.2020, author = {Oates, Chris and Cockayne, Jon and Prangle, Dennis and Sullivan, T. J. and Girolami, Mark}, title = {Optimality criteria for probabilistic numerical methods}, volume = {27}, journal = {Multivariate Algorithms and Information-Based Complexity}, editor = {Hickernell, F. J. and Kritzer, P.}, publisher = {De Gruyter}, arxiv = {http://arxiv.org/abs/1901.04326}, doi = {10.1515/9783110635461-005}, pages = {65 -- 88}, year = {2020}, abstract = {It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that the decision-theoretic framework is neither appropriate nor sufficient. To this end, we consider an alternative optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an average-case-optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes in which optimal probabilistic numerical methods can be developed.}, language = {en} } @article{KerstingSullivanHennig2020, author = {Kersting, Hans and Sullivan, T. J. and Hennig, Philipp}, title = {Convergence rates of Gaussian ODE filters}, volume = {30}, journal = {Statistics and Computing}, publisher = {Springer}, address = {Statistics and Computing}, arxiv = {http://arxiv.org/abs/1807.09737}, doi = {10.1007/s11222-020-09972-4}, pages = {1791 -- 1816}, year = {2020}, language = {en} } @article{LieSullivanTeckentrup2021, author = {Lie, Han Cheng and Sullivan, T. J. and Teckentrup, Aretha}, title = {Error bounds for some approximate posterior measures in Bayesian inference}, journal = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, publisher = {Springer}, arxiv = {http://arxiv.org/abs/1911.05669}, doi = {10.1007/978-3-030-55874-1_26}, pages = {275 -- 283}, year = {2021}, language = {en} } @misc{LieSullivan2016, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Cameron--Martin theorems for sequences of Cauchy-distributed random variables}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60230}, year = {2016}, abstract = {Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.}, language = {en} } @misc{Sullivan2016, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors}, issn = {1438-0064}, doi = {10.3934/ipi.2017040}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59422}, year = {2016}, abstract = {This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451-559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen-Lo{\`e}ve expansion for square-integrable random variables can be used to sample such measures. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.}, language = {en} } @misc{CockayneOatesSullivanetal.2016, author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59513}, year = {2016}, abstract = {This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions.}, language = {en} } @article{LieSullivan2016, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Cameron--Martin theorems for sequences of Cauchy-distributed random variables}, journal = {arXiv}, arxiv = {http://arxiv.org/abs/1608.03784}, pages = {1608.03784}, year = {2016}, abstract = {Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.}, language = {en} } @misc{NavaYazdaniHegevonTycowiczetal.2018, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and von Tycowicz, Christoph and Sullivan, T. J.}, title = {A Shape Trajectories Approach to Longitudinal Statistical Analysis}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69759}, year = {2018}, abstract = {For Kendall's shape space we determine analytically Jacobi fields and parallel transport, and compute geodesic regression. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As application example we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data only.}, language = {en} } @inproceedings{TeymurLieSullivanetal.2018, author = {Teymur, Onur and Lie, Han Cheng and Sullivan, T. J. and Calderhead, Ben}, title = {Implicit probabilistic integrators for ODEs}, booktitle = {Advances in Neural Information Processing Systems 31 (NIPS 2018)}, arxiv = {http://arxiv.org/abs/1805.07970}, year = {2018}, language = {en} } @article{KlebanovSchusterSullivan2020, author = {Klebanov, Ilja and Schuster, Ingmar and Sullivan, T. J.}, title = {A rigorous theory of conditional mean embeddings}, volume = {2}, journal = {SIAM Journal on Mathematics of Data Science}, number = {3}, arxiv = {http://arxiv.org/abs/1912.00671}, doi = {10.1137/19M1305069}, pages = {583 -- 606}, year = {2020}, language = {en} } @article{GirolamiIpsenOatesetal.2019, author = {Girolami, Mark A. and Ipsen, Ilse C. F. and Oates, Chris and Owen, Art B. and Sullivan, T. J.}, title = {Editorial: Special edition on probabilistic numerics}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {doi:10.1007/s11222-019-09892-y}, pages = {1181 -- 1183}, year = {2019}, language = {en} } @article{NavaYazdaniHegeSullivanetal.2020, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and Sullivan, T. J. and von Tycowicz, Christoph}, title = {Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications}, volume = {62}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, arxiv = {http://arxiv.org/abs/1906.11950}, doi = {10.1007/s10851-020-00945-w}, pages = {549 -- 559}, year = {2020}, abstract = {We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.}, language = {en} } @article{KlebanovSprungkSullivan2021, author = {Klebanov, Ilja and Sprungk, Bj{\"o}rn and Sullivan, T. J.}, title = {The linear conditional expectation in Hilbert space}, volume = {27}, journal = {Bernoulli}, number = {4}, arxiv = {http://arxiv.org/abs/2008.12070}, doi = {10.3150/20-BEJ1308}, pages = {2299 -- 2299}, year = {2021}, language = {en} } @article{BonnetAkianSavinetal.2020, author = {Bonnet, Luc and Akian, Jean-Luc and Savin, {\´E}ric and Sullivan, T. J.}, title = {Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification}, volume = {13}, journal = {Algorithms}, number = {8}, arxiv = {http://arxiv.org/abs/2007.05236}, doi = {10.3390/a13080196}, pages = {196}, year = {2020}, abstract = {Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design.}, language = {en} } @inproceedings{Sullivan2017, author = {Sullivan, T. J.}, title = {Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors}, volume = {17}, booktitle = {88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017}, number = {1}, arxiv = {http://arxiv.org/abs/1710.05610}, doi = {10.1002/pamm.201710402}, pages = {871 -- 874}, year = {2017}, language = {en} } @article{Sullivan2017, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors}, volume = {11}, journal = {Inverse Problems and Imaging}, number = {5}, arxiv = {http://arxiv.org/abs/1605.05898}, doi = {10.3934/ipi.2017040}, pages = {857 -- 874}, year = {2017}, language = {en} } @article{LieSullivanStuart2019, author = {Lie, Han Cheng and Sullivan, T. J. and Stuart, Andrew}, title = {Strong convergence rates of probabilistic integrators for ordinary differential equations}, volume = {29}, journal = {Statistics and Computing}, number = {6}, arxiv = {http://arxiv.org/abs/1703.03680}, doi = {10.1007/s11222-019-09898-6}, pages = {1265 -- 1283}, year = {2019}, abstract = {Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator.}, language = {en} } @inproceedings{CockayneOatesSullivanetal.2017, author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems}, volume = {1853}, booktitle = {Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering}, arxiv = {http://arxiv.org/abs/1701.04006}, doi = {10.1063/1.4985359}, pages = {060001-1 -- 060001-8}, year = {2017}, abstract = {This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model}, language = {en} }